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Does it make sense to ask about cardinality of closed finite topology defined on the set of Natural Numbers? Is it countable or uncountable? Is it possible to prove that it is countable?

Further, if it countable, is every closed finite topology (cofinite topology) defined on countable set again countable?

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up vote 4 down vote accepted

First note that the map $X\mapsto\Bbb N\setminus X$ is a bijection between the non-trivial cofinite open sets, and the finite open sets.

Next show that $\{A\subseteq\Bbb N\mid A\text{ is finite}\}$ is a countably infinite set.

Lastly note that if $|X|=|Y|$ then defining the cofinite topology on $X$ would be easily transferable to $Y$. Hint: Every a bijection is a homeomorphism.

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Beat me by a split second, but then this is your home turf! Allow me to quote your own answer here… to the fact that (using choice, which is not needed here) the cardinality of the set of finite subsets of an infinite set $A$ is the same as the cardinality of $A$. – Andreas Caranti Mar 23 '13 at 12:08

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